Topological K-theory is a powerful invariant of spaces. For instance, in the ’60s Adams and Atiyah used it to give a simple proof of the Hopf invariant one problem. In the ’80s and ’90s homotopy theorists proved that canonical generalizations of K-theory exist after completion at a fixed prime. We will begin with an intuitive introduction to topological K-theory. After this, we will explain how computations with these generalizations of K-theory relate to arithemetic geometry and are compatible with the structure observed on certain quantum field theories.